Gravesande, Willem Jacob 's - An essay on perspective

Now we have proved, that F G is the half of
F B, therefore G N is likewiſe equal to the half
of B L, and conſequently equal to the Height
of the ſuppoſed Perpendicular.

Again, the ſimilar Triangles F G N and F a I
give

F G : F a : : G N : a I.

But F G : F a : : G D : a H; becauſe the Tri-
angles F G D and F a H are ſimilar.

Whence

G N : a I : : G D : a H.

Now becauſe G N has been proved to be e-
qual to the Perpendicular, whoſe Perſpective is
requir’d and D G is ſuppoſed equal to that Per-
pendicular; it follows, that G N and G D are
equal; and therefore a I and a H are alſo equal.
Q E D.

Scholium.

I might have aſſumed C P equal to the Perpen-
dicular, and uſed the Points C and P inſtead of
B and L. But uſing the ſaid Points B and L is
better: For when the Points C and P are uſed,
the Horizontal Line muſt almoſt always be con-
tinued, that ſo a Line drawn through the Points
c and a may cut it; moreover this Interſection
will ſometimes be at an infinite Diſtance; where-
as in uſing the Point B, M N can never be
greater than thrice the Breadth of the Deſign to
be drawn.

Corollary.

The ſixth Problem may be ſolv’d by this;
for a Point elevated above the Geometrical
Plane, may be conceived as the Extremity of a
Perpendicular to the Geometrical Plane.

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